set one mesh as constraint for another
Kirbinator is on the right track. The "correct" way to do this is either to have them both in the same object, and deformed (animated) by the same armature, or to have two armatures that are both controlled by a third (you move one thing, and it moves both others).

You can just select both objects and press ctrlj to join them into one mesh, but that can be a little tricky. Experimenting with it wouldn't be hard.

Where you are now, though, it seems like the best solution would be a shrinkwrap modifier. You'll need to experiment with whether to shrinkwrap the lungs to the inside of the chest, or the chest to the outside of the lungs, but one of those should work.

The reason I mention this is that the shrinkwrap modifier has a setting for "Offset" which will shrinkwrap to a certain distance from the target, instead of to the target itself. So, for example, you could shrinkwrap the chest to the lungs, but with a 1cm gap.

IIRC, you said the lungs had the armature, in which case you'd want to shrinkwrap the chest to match the lungs. Either way, you'll want to shrinkwrap the one that is NOT rigged onto the one that is.

Hope that helps!